Confluent Hypergeometric Functions Research Articles

Analytical expression for continuum–continuum transition amplitude of hydrogen-like atoms with angular-momentum dependence

Abstract Attosecond chronoscopy typically utilises interfering two-photon transitions to access the phase information. Simulating these two-photon transitions is challenging due to the continuum–continuum transition term. The hydrogenic approximation within second-order perturbation theory has been widely used due to the existence of analytical expressions of the wave functions. So far, only (partially) asymptotic results have been derived, which fail to correctly describe the low-kinetic-energy behaviour, especially for high angular-momentum states. Here, we report an analytical expression that overcomes these limitations. It is based on the Appell’s F 1 function and uses the confluent hypergeometric function of the second kind as the intermediate state. We show that the derived formula quantitatively agrees with the numerical simulations using the time-dependent Schrödinger equation for various angular-momentum states, which improves the accuracy compared to the other analytical approaches that were previously reported. Furthermore, we give an angular-momentum-dependent asymptotic form of the outgoing wavefunction and the corresponding continuum–continuum dipole transition amplitudes.

Bivariate Pareto–Feller Distribution Based on Appell Hypergeometric Function

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown.

Analysis of the beta-logarithmic function and its properties

Abstract In this article, we aim to present a new extension of the beta-logarithmic function with the help of an extended beta function and logarithmic mean. Then we introduce its statistical properties and investigate some basic important formulas of extended beta-logarithmic functions, such as integral representation, summation formula and Mellin transform. Further, we consider a new type of hypergeometric and confluent hypergeometric functions using this extended beta-logarithmic function and discuss their various result.

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The Mittag-Leffler and confluent hypergeometric functions were originally developed to extend the exponential function and its area of applications. This study aims to examine some operators involving generalized Mittag-Leffler-type functions in the kernels, employing the generalized Fox-Wright function in specific circumstances. Furthermore, we investigate some of the commonly utilized generalized fractional integral operators in fractional calculus. Moreover, a numerical technique is developed to solve fractional differential equations of both kinds, linear and nonlinear. The graphic results of the examples show how effective this method is at solving fractional differential equations. Lastly, various effects and implications of these results are thoroughly examined.

Inequalities for Basic Special Functions Using Hölder Inequality

Let p,q≥1 be two real numbers such that 1p+1q=1, and let a,b∈R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form |f(ap+bq)|≤|f(a)|1p|f(b)|1q, and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

Analytical solutions for the advective–diffusive ice column in the presence of strain heating

Abstract. A thorough understanding of ice thermodynamics is essential for an accurate description of glaciers, ice sheets and ice shelves. Yet there exists a significant gap in our theoretical knowledge of the time-dependent behaviour of ice temperatures due to the inevitable compromise between mathematical tractability and the accurate description of physical phenomena. In order to bridge this shortfall, we have analytically solved the 1D time-dependent advective–diffusive heat problem including additional terms due to strain heating and depth-integrated horizontal advection. Newton's law of cooling is applied as a Robin-type top boundary condition to consider potential non-equilibrium temperature states across the ice–air interface. The solution is expressed in terms of confluent hypergeometric functions following a separation of variables approach. Non-dimensionalization reduces the parameter space to four numbers that fully determine the shape of the solution at equilibrium: surface insulation, effective geothermal heat flow, the Péclet number and the Brinkman number. The initial temperature distribution exponentially converges to the stationary solution. Transient decay timescales are only dependent on the Péclet number and the surface insulation, so higher advection rates and lower insulating values imply shorter equilibration timescales, respectively. In contrast, equilibrium temperature profiles are mostly independent of the surface insulation parameter. We have extended our study to a broader range of vertical velocities by using a general power-law dependence on depth, unlike prior studies limited to linear and quadratic velocity profiles. Lastly, we present a suite of benchmark experiments to test numerical solvers. Four experiments of gradually increasing complexity capture the main physical processes for heat propagation. Analytical solutions are then compared to their numerical counterparts upon discretization over unevenly spaced coordinate systems. We find that a symmetric scheme for the advective term and a three-point asymmetric scheme for the basal boundary condition best match our analytical solutions. A further convergence study shows that n≥15 vertical points are sufficient to accurately reproduce the temperature profile. The solutions presented herein are general and fully applicable to any problem with an equivalent set of boundary conditions and any given initial temperature distribution.

On Properties of the Hyperbolic Distribution

This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Existence of quantum states for Klein–Gordon particles based on exact and approximate scenarios

In the present study, Kummer’s eigenvalue spectra from a charged spinless particle located at spherical pseudo-dot of the form r 2 + 1/r 2 is reported. Here, it is shown how confluent hypergeometric functions have principal quantum numbers for considered spatial confinement. To study systematically both constant rest-mass, m 0 c 2 and spatial-varying mass of the radial distribution m 0 c 2 + S(r), the Klein–Gordon equation is solved under exact case and approximate scenario for a constant mass and variable usage, respectively. The findings related to the relativistic eigenvalues of the Klein–Gordon particle moving in the spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than m 0 c 2 = 1 fm−1 in exact scenario. Following analysis also shows eigenvalues satisfy the range of E < m 0 c 2 through approximate scenario.

Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction

A significance of Marangoni-driven convective magnetohydrodynamics (MHD) was investigated in a planned study. The present study examines the impact of radiation and chemical reactions on the two dimensional boundary layer flow of bi-viscous Bingham fluid on a thermosolutal Marangoni boundary with magnetic field and heat source/sink. The physical flow problem is mathematically modelled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The elegant analytical method is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The analytical solution for the temperature and concentration profiles is expressed in terms of Kummer's confluent hypergeometric function. The features of flow characteristics such as velocity, temperature, and concentration profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation through graphs. Surface tension is considered to be corresponding to heat and mass. The results reveal that the velocity profile decreases with increasing the magnetic field, while the Marangoni number value increases with increasing the velocity profile. Moreover, the temperature profile rises with rising the radiation parameter and whereas the value chemical reaction parameter upsurges with decline concentration profile. The present problem has great interest due to its applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, crystal growth in space, biological fluids, blood, synovial liquids, plasma, lubricants, many pharmaceutical products, proteins, sewage sludge, china clay, and many more.

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities

The principal aim of this paper is to introduce the extended Voigt-type function Vμ,ν(x,y) and its counterpart extension Wμ,ν(x,y), involving the Neumann function Yν in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions K(x,y) and L(x,y), and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions Vμ,ν(r)(x,y),Wμ,ν(r)(x,y) and the r positive integer of the initial extensions Vμ,ν(x,y),Wμ,ν(x,y). Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions Ψ2(r). Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind Qη−ν in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for Vμ,ν(x,y) and Wμ,ν(x,y). Particularly interesting results are presented for the Neumann function Yν and for the Struve Hν function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks.

Частица Штюкельберга в электрическом поле, решения с цилиндрической симметрией

In the present paper, the system of 11 equations for massive Stueckelberg particle is studied in presence of the external uniform electric field. We apply covariant formalism according to the general tetrad approach by Tetrode-Weyl-Fock-Ivanenko specified for cylindrical coordinates. After separating the variables, we derive the system of the first-order differential equations in partial derivatives with respect to coordinates (r, z). To resolve this system, we apply the Fedorov- Gronskiy method, thereby we consider the 11-dimensional spin operator and find on this base three projective operators, which permit us to expand the complete wave function in the sum of three parts. Besides, according to the general method, dependence of each projective constituent on the variable r should be determined by only one function. Also, in accordance with the general method we impose the first-order constraints which permit us to transform all differential equations in partial derivatives with respect to coordinates (r, z) into the system of 11 first-order ordinary differential equations in the variable z. The last system is solved in terms of confluent hypergeometric functions. In total, four independent types of solutions have been constructed, in contrast to the case of the ordinary spin 1 particle described by Daffin- Kemmer equation when only three types of solutions are possible.

On the analytic extension of three ratios of Horn's confluent hypergeometric function $\mathrm{H}_7$

In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $\mathrm{H}_7$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.

An analytical study on the influence of magnetic field-dependent viscosity on viscous and ohmic dissipative second-grade fluid boundary layers

This article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Bounds for novel extended beta and hypergeometric functions and related results

We introduce a new unified extension of the integral form of Euler’s beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer’s confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function.

Fractional calculus of modified special functions involving the generalized M-series in their kernels and illustrative examples

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.